Obtain a coordinate-induced basis for the tangent space and
cotangent space at points of a differentiable manifold, construct
a coordinate induced basis for arbitrary tensors and obtain the
components of tensors in this basis;

Determine whether a particular map is a tensor by either checking
multi-linearity or by showing that the components transform
according to the tensor transformation law;

Construct manifestly chart-free definitions of the Lie derivative
of a function and a vector, to compute these derivatives in a
particular chart and hence compute the Lie derivative of an
arbitrary tensor;

Obtain an expression for the Riemann curvature tensor in an
arbitrary basis for a manifold with vanishing torsion, provide a
geometric interpretation of what this tensor measures, derive
various symmetries and results involving the curvature tensor;

Define the metric, the Levi-Civita connection and the metric
curvature tensor and compute the components of each of these
tensors given a particular line-element;